Extremely Hard Question

Question

Points  and  lie in a plane with .  How many locations for point  in this plane are there such that the triangle with vertices , , and  is a right triangle with area   square units?

(b) If area is 25

(c)If area is 30

Answer 1

The problem states that line segment PQ is 10 units long. Imagine a horizontal plane containing PQ, with point P to the left and point Q to the right.

(a) Consider these 8 different locations for point R in this plane. For each of these 8 locations, the right triangle PQR has an area of 15 square units.

1. Line segment PR is 3 units long and perpendicular to PQ at point P. Point R is in front of point P.
2. Line segment PR is 3 units long and perpendicular to PQ at point P. Point R is in back of point P.
3. Line segment QR is 3 units long and perpendicular to PQ at point Q. Point R is in front of point Q.
4. Line segment QR is 3 units long and perpendicular to PQ at point Q. Point R is in back of point Q.
5. Line segments PR and QR are perpendicular at point R. R is on a semicircle with a 5 unit radius in front of line segment PQ, where points P and Q are the endpoints of this semicircle. The perpendicular distance from point R to line segment PQ is 3 units. PR is shorter than QR.
6. Line segments PR and QR are perpendicular at point R. R is on a semicircle with a 5 unit radius in front of line segment PQ, where points P and Q are the endpoints of this semicircle. The perpendicular distance from point R to line segment PQ is 3 units. PR is longer than QR.
7. Line segments PR and QR are perpendicular at point R. R is on a semicircle with a 5 unit radius in back of line segment PQ, where points P and Q are the endpoints of this semicircle. The perpendicular distance from point R to line segment PQ is 3 units. PR is shorter than QR.
8. Line segments PR and QR are perpendicular at point R. R is on a semicircle with a 5 unit radius in back of line segment PQ, where points P and Q are the endpoints of this semicircle. The perpendicular distance from point R to line segment PQ is 3 units. PR is longer than QR.

(b) Consider these 6 different locations for point R in this plane. For each of these 6 locations, the right triangle PQR has an area of 25 square units.

1. Line segment PR is 5 units long and perpendicular to PQ at point P. Point R is in front of point P.
2. Line segment PR is 5 units long and perpendicular to PQ at point P. Point R is in back of point P.
3. Line segment QR is 5 units long and perpendicular to PQ at point Q. Point R is in front of point Q.
4. Line segment QR is 5 units long and perpendicular to PQ at point Q. Point R is in back of point Q.
5. Line segments PR and QR are perpendicular at point R. R is on a semicircle with a 5 unit radius in front of line segment PQ, where points P and Q are the endpoints of this semicircle. The perpendicular distance from point R to line segment PQ is 5 units. PR is equal in length to QR.
6. Line segments PR and QR are perpendicular at point R. R is on a semicircle with a 5 unit radius in back of line segment PQ, where points P and Q are the endpoints of this semicircle. The perpendicular distance from point R to line segment PQ is 5 units. PR is equal in length to QR.

(c) Consider these 4 different locations for point R in this plane. For each of these 4 locations, the right triangle PQR has an area of 30 square units.

1. Line segment PR is 6 units long and perpendicular to PQ at point P. Point R is in front of point P.
2. Line segment PR is 6 units long and perpendicular to PQ at point P. Point R is in back of point P.
3. Line segment QR is 6 units long and perpendicular to PQ at point Q. Point R is in front of point Q.
4. Line segment QR is 6 units long and perpendicular to PQ at point Q. Point R is in back of point Q.

Note that there is no point on the semicircle with endpoints P and Q that is 6 units away from line segment PQ, so there is no right triangle, with R as the right angle point, that can have an area of 30 square units.